A note on eigenvalue computation for a tridiagonal matrix with real eigenvalues

The target matrix of the dhLV algorithm is already shown to be a class of nonsymmetric band matrix with complex eigenvalues. In the case where the band width M=1 in the dhLV algorithm, it is applicable to a tridiagonal matrix, with real eigenvalues, whose upper and lower subdiagonal entries are restricted to be positive and 1, respectively. In this paper, we first clarify that the dhLV algorithm is also applicable to the eigenvalue computation of nonsymmetric tridiagonal matrix with relaxing the restrictions for subdiagonal entries. We also demonstrate that the wellknown packages are not always desirable for computing nonsymmetric eigenvalues with respect to numerical accuracy. Through some numerical examples, it is shown that the tridiagonal eigenvalues computed by the dhLV algorithm are to high relative accuracy.